On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles

For \begin{document}$ C^2 $\end{document} cos-type potentials, large coupling constants, and fixed \begin{document}$ Diophantine $\end{document} frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point \begin{document}$ E $\end{document}, \begin{document}$ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $\end{document}, where \begin{document}$ \beta\in [\frac{1}{2},1] $\end{document}. Our approach is a further improvement on the papers [15] and [17].